Cell Complexities in Hyperplane Arrangements

We derive improved bounds on the complexity, i.e., the total number of faces of all dimensions, of many cells in arrangements of hyperplanes in higher dimensions, and use these bounds to obtain a very simple proof of an earlier bound, due to Aronov, Matousek, and Sharir, on the sum of squares of cell complexities in such an arrangement.     

Data-sparse approximation to the operator-valued functions of elliptic operator

In previous papers the arithmetic of hierarchical matrices has been described, which allows us to compute the inverse, for instance, of finite element stiffness matrices discretising an elliptic operator The required computing time is up to logarithmic factors linear in the dimension of the matrix. In particular, this technique can be used for the computation of the discrete analogue of a resolvent

In the present paper, we consider various operator functions, the operator exponential negative fractional powers , the cosine operator function and, finally, the solution operator of the Lyapunov equation. Using the Dunford-Cauchy representation, we get integrals which can be discretised by a quadrature formula which involves the resolvents mentioned above. We give error estimates which are partly exponentially, partly polynomially decreasing.

     

Uniqueness for an elliptic-parabolic problem with Neumann boundary condition

We consider the problem in a smooth boundary domain , as well as the corresponding evolution equation . For the stationary equation we show existence results, then we adapt the techniques of doubling of variables to the case of the homogeneous Neumann boundary conditions and obtain the appropriate L ¹ -contraction principle and uniqueness. Subsequently, we are able to apply the nonlinear semigroup theory and prove the L ¹ -contraction principle for the associated evolution equation.     

Singular integrals generated by zonal measures

We study -mapping properties of the rough singular integral operator depending on a finite Borel measure on the unit sphere in . It is shown that the conditions , imply the -boundedness of for all provided that 2$"> and is zonal.

     

Deterministic primality test for numbers of the form odd

We use a result of E. Lehmer in cubic residuacity to find an algorithm to determine primality of numbers of the form , odd, . The algorithm represents an improvement over the more general algorithm that determines primality of numbers of the form , , presented by Berrizbeitia and Berry (1999).

     

Helgason-Marchaud inversion formulas for Radon transforms

Let be either the hyperbolic space or the unit sphere , and let be the set of all -dimensional totally geodesic submanifolds of . For and , the totally geodesic Radon transform is studied. By averaging over all at a distance from , and applying Riemann-Liouville fractional differentiation in , S. Helgason has recovered . We show that in the hyperbolic case this method blows up if does not decrease sufficiently fast. The situation can be saved if one employs Marchaud's fractional derivatives instead of the Riemann-Liouville ones. New inversion formulas for , are obtained.

     

Wigner's theorem in Hilbert -algebras of compact operators

Let be a Hilbert -module over the -algebra of all compact operators on a Hilbert space. It is proved that any function which preserves the absolute value of the -valued inner product is of the form , where is a phase function and is an -linear isometry. The result generalizes Molnár's extension of Wigner's classical unitary-antiunitary theorem.

     

A blood vessel exposed to ultrasound: a mathematical simulation of the temperature field

In this article we present a mathematical simulation of the temperature field in and around a blood vessel when it is sonicated by a focused ultrasound beam. A simplified geometry is considered: a cylindrical blood vessel is embedded in tissue parallel to a flat skin surface. The ultrasound transducer is placed on the skin above the blood vessel, perpendicular to the skin surface. The 3D geometry of the problem is simplified by transformation, which maps the domain into a parallelepiped. A computational algorithm and computer program were developed. The simulation provides the conditions for successful occlusion of a blood vessel and demonstrates the significant role of the blood flow rate on the temperature difference between the vessel wall and the surrounding tissue. Comparing the predictions with published experimental data tested the validity of the method.     

Josephson plasma resonance and phonon anomalies in trilayer Bi2Sr2Ca2Cu3O10

The far-infrared (FIR) c axis conductivity of a Bi2223 crystal has been measured by ellipsometry. Below T(c) a strong absorption band develops near 500 cm(-1), corresponding to a transverse Josephson plasmon. The related increase in FIR spectral weight leads to a giant violation of the Ferrell-Glover-Tinkham sum rule. The gain in c axis kinetic energy accounts for a sizable part of the condensation energy. We also observe phonon anomalies which suggest that the Josephson currents lead to a drastic variation of the local electric field within the block of closely spaced CuO2 planes.     

Data storage in optical fibers and reconstruction by use of low-coherence spectral interferometry

We demonstrate optical data storage in optical fibers and reconstruction by use of low-coherence spectral interferometry. The information was stored by means of writing fiber Bragg gratings with different central wavelengths at different locations of the fiber. We need only a single short pulse is needed to read all the stored data. The maximum theoretical reconstruction rate that can be obtained with our technique is 10 Tbits/s. Our storage technique can be useful for identifying users in optical communication networks.